Theorem dcomex 10439

Description: The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)

Assertion

Ref	Expression
dcomex	⊢ ω ∈ V

Proof of Theorem dcomex

Dummy variables 𝑡 𝑠 𝑥 𝑓 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 8485	. . . . . . 7 ⊢ 1o ≠ ∅
2		df-br 5149	. . . . . . . 8 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩})
3		elsni 4645	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩)
4		fvex 6902	. . . . . . . . . 10 ⊢ (𝑓‘𝑛) ∈ V
5		fvex 6902	. . . . . . . . . 10 ⊢ (𝑓‘suc 𝑛) ∈ V
6	4, 5	opth1 5475	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩ → (𝑓‘𝑛) = 1o)
7	3, 6	syl 17	. . . . . . . 8 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → (𝑓‘𝑛) = 1o)
8	2, 7	sylbi 216	. . . . . . 7 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → (𝑓‘𝑛) = 1o)
9		tz6.12i 6917	. . . . . . 7 ⊢ (1o ≠ ∅ → ((𝑓‘𝑛) = 1o → 𝑛𝑓1o))
10	1, 8, 9	mpsyl 68	. . . . . 6 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1o)
11		vex 3479	. . . . . . 7 ⊢ 𝑛 ∈ V
12		1oex 8473	. . . . . . 7 ⊢ 1o ∈ V
13	11, 12	breldm 5907	. . . . . 6 ⊢ (𝑛𝑓1o → 𝑛 ∈ dom 𝑓)
14	10, 13	syl 17	. . . . 5 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
15	14	ralimi 3084	. . . 4 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
16		dfss3 3970	. . . 4 ⊢ (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
17	15, 16	sylibr 233	. . 3 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
18		vex 3479	. . . . 5 ⊢ 𝑓 ∈ V
19	18	dmex 7899	. . . 4 ⊢ dom 𝑓 ∈ V
20	19	ssex 5321	. . 3 ⊢ (ω ⊆ dom 𝑓 → ω ∈ V)
21	17, 20	syl 17	. 2 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ∈ V)
22		snex 5431	. . 3 ⊢ {⟨1o, 1o⟩} ∈ V
23	12, 12	fvsn 7176	. . . . . . . 8 ⊢ ({⟨1o, 1o⟩}‘1o) = 1o
24	12, 12	funsn 6599	. . . . . . . . 9 ⊢ Fun {⟨1o, 1o⟩}
25	12	snid 4664	. . . . . . . . . 10 ⊢ 1o ∈ {1o}
26	12	dmsnop 6213	. . . . . . . . . 10 ⊢ dom {⟨1o, 1o⟩} = {1o}
27	25, 26	eleqtrri 2833	. . . . . . . . 9 ⊢ 1o ∈ dom {⟨1o, 1o⟩}
28		funbrfvb 6944	. . . . . . . . 9 ⊢ ((Fun {⟨1o, 1o⟩} ∧ 1o ∈ dom {⟨1o, 1o⟩}) → (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o))
29	24, 27, 28	mp2an 691	. . . . . . . 8 ⊢ (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o)
30	23, 29	mpbi 229	. . . . . . 7 ⊢ 1o{⟨1o, 1o⟩}1o
31		breq12 5153	. . . . . . . 8 ⊢ ((𝑠 = 1o ∧ 𝑡 = 1o) → (𝑠{⟨1o, 1o⟩}𝑡 ↔ 1o{⟨1o, 1o⟩}1o))
32	12, 12, 31	spc2ev 3598	. . . . . . 7 ⊢ (1o{⟨1o, 1o⟩}1o → ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡)
33	30, 32	ax-mp 5	. . . . . 6 ⊢ ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡
34		breq 5150	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → (𝑠𝑥𝑡 ↔ 𝑠{⟨1o, 1o⟩}𝑡))
35	34	2exbidv 1928	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∃𝑠∃𝑡 𝑠𝑥𝑡 ↔ ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡))
36	33, 35	mpbiri 258	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → ∃𝑠∃𝑡 𝑠𝑥𝑡)
37		ssid 4004	. . . . . . 7 ⊢ {1o} ⊆ {1o}
38	12	rnsnop 6221	. . . . . . 7 ⊢ ran {⟨1o, 1o⟩} = {1o}
39	37, 38, 26	3sstr4i 4025	. . . . . 6 ⊢ ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}
40		rneq 5934	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 = ran {⟨1o, 1o⟩})
41		dmeq 5902	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → dom 𝑥 = dom {⟨1o, 1o⟩})
42	40, 41	sseq12d 4015	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}))
43	39, 42	mpbiri 258	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 ⊆ dom 𝑥)
44		pm5.5 362	. . . . 5 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
45	36, 43, 44	syl2anc 585	. . . 4 ⊢ (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
46		breq 5150	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → ((𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
47	46	ralbidv 3178	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
48	47	exbidv 1925	. . . 4 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
49	45, 48	bitrd 279	. . 3 ⊢ (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
50		ax-dc 10438	. . 3 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
51	22, 49, 50	vtocl 3550	. 2 ⊢ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)
52	21, 51	exlimiiv 1935	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  Vcvv 3475   ⊆ wss 3948  ∅c0 4322  {csn 4628  ⟨cop 4634   class class class wbr 5148  dom cdm 5676  ran crn 5677  suc csuc 6364  Fun wfun 6535  ‘cfv 6541  ωcom 7852  1_oc1o 8456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722  ax-dc 10438

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-fv 6549  df-1o 8463

This theorem is referenced by:  axdc2lem  10440  axdc3lem  10442  axdc4lem  10447  axcclem  10449  precsexlem10  27652